Note 4.2.1. From now on we will make an intensive use of the properties of C0-
semigroups in a nuclear space and its strong dual. For a review of the relevant facts about the theory of C0-semigroups in locally convex spaces the reader is referred to
Appendix D.
In this section we will introduce the general model of stochastic evolution equations in the dual of a nuclear space driven by a nuclear cylindrical martingale-valued measure. Let Φ be a locally convex space and Ψ be a quasi-complete, bornological, nuclear space, both defined over R. Let U be a topological space. We are concerned with the following class of stochastic evolution equations
dXt= (A0Xt+ B(t, Xt))dt +
Z
U
F (t, u, Xt)M (dt, du), for t ≥ 0, (4.10)
where we will assume the following: Assumption 4.2.2.
(A1) A is the infinitesimal generator of a (C0, 1) -semi-group {S(t)}t≥0 on Ψ .
(A2) M is a nuclear cylindrical martingale-valued measure on R+× R, where R is
a ring R ⊆ B(U ) that generates the Borel σ -algebra B(U ) of the topological space U , and the covariance of M is determined by the measure λ = Leb on R+, a σ -finite
Borel measure µ on U , and the semi-norms {qr,u : r ∈ R+, u ∈ U } ; all satisfying the
conditions in Definition 3.1.3 and Assumption 3.1.9.
(A3) B : R+× Ψ0 → Ψ0 is such that the map (r, g) 7→ B(r, g)[ψ] is B(R+) ⊗ B(Ψ0β) -
measurable, for every ψ ∈ Ψ .
(A4) F = {F (r, u, g) : r ∈ R+, u ∈ U, g ∈ Ψ0} is such that
(a) F (r, u, g) ∈ L(Φ0qr,u, Ψ0β) , ∀r ≥ 0 , u ∈ U , g ∈ Ψ0.
(b) The mapping (r, u, g) 7→ qr,u(F (r, u, g)0φ, ψ) is B(R+)⊗B(U )⊗B(Ψ0β) -measurable,
for every φ ∈ Φ , ψ ∈ Ψ .
Note that Ψ being reflexive (Theorem 1.1.7(2)), assumption (A1) implies that A0 (the dual operator of A ) is the infinitesimal generator of the dual semi-group {S(t)0}t≥0
and this last is a C0-semigroup on Ψ0β (see Theorem D.2.7).
Remark 4.2.3. It is well known that the solutions of stochastic evolutions equations in infinite dimensional spaces are not in general c`adl`ag, for that reason instead of considering equations with left limits in the right hand side of (4.10) we require only
4.2. Stochastic Evolution Equations: The General Setting 89
that our solution be predictable (see Definitions 4.2.5 and 4.2.7). For a more detailed discussion on this the reader is referred to Section 9.2.1 of Peszat and Zabczyk [85]. Remark 4.2.4. The use of (C0, 1) -semi-groups for the study of stochastic evolution
equations in duals of nuclear spaces has its origins in the work of Kallianpur and P´erez- Abreu [52] where they considered such semigroups on a nuclear Fr´echet space. Indeed, the authors considered the more general context of (C0, 1) -reversed evolution systems.
Again in the framework of nuclear Fr´echet spaces, Ding [26] also used (C0, 1) -semi-
groups to study stochastic evolution equations. He assumed that the dual semigroup is (C0, 1) , with a more restrictive hypothesis that there exists a family of Hilbertian
semi-norms generating the nuclear topology on Ψ0β such that these semi-norms satisfy the conditions of Theorem D.2.4.
We are interested in to studying weak and mild solutions to (4.10). The precise formu- lation of these types of solutions is given below.
Definition 4.2.5. A Ψ0β-valued regular and predictable process X = {Xt}t≥0 is called
a weak solution to (4.10) if
(a) For every t > 0 , X , B and F satisfy the following conditions: P ω ∈ Ω : Z t 0 |Xr(ω)[ψ]| dr < ∞ = 1, ∀ ψ ∈ Ψ. P ω ∈ Ω : Z t 0 |B(r, Xr(ω))[ψ]| dr < ∞ = 1, ∀ ψ ∈ Ψ. P ω ∈ Ω : Z t 0 Z U qr,u(F (r, u, Xr(ω))0ψ)2µ(du)dr < ∞ = 1, ∀ ψ ∈ Ψ. (b) For every ψ ∈ Dom(A) and every t ≥ 0 , P-a.e.
Xt[ψ] = X0[ψ] + Z t 0 (Xr[Aψ] + B(r, Xr)[ψ])dr (4.11) + Z t 0 Z U F (r, u, Xr)0ψM (dr, du),
where the first integral in the right-hand side of (4.11) is a Lebesgue integral that is defined for each ψ ∈ Ψ for P-a.e. ω ∈ Ω. The second integral in the right-hand side of (4.11) is the weak stochastic integral of F0ψ = {F (r, u, Xr(ω))0ψ : r ∈
[0, t], ω ∈ Ω, u ∈ U } ∈ Λ2,locw (t) , and is well-defined for all ψ ∈ Ψ .
Proposition 4.2.6. The assumptions (A1)-(A4) together with the conditions (a) of Definition 4.2.5 are sufficient to guarantee the existence of all the integrals in (4.11). Proof. We start with the deterministic integral. Fix ψ ∈ Ψ . The fact that X is pre- dictable together with (A3), implies that the map (r, ω) 7→ (Xr(ω)[Aψ]+B(r, Xr(ω))[ψ])
is P∞-measurable. Then, condition (a) of Definition 4.2.5 implies that the above map
is Lebesgue integrable on [0, ∞) for P-a.e. ω ∈ Ω.
Now we prove that the stochastic integral is well-defined. To do this, fix ψ ∈ Ψ . Then, the fact that X is predictable together with (A4) implies that F (r, u, Xr)0ψ ∈ Φ0qr,u,
for each r ≥ 0 , ω ∈ Ω and u ∈ U , and that the map (r, ω, u) 7→ qr,u(F (r, u, Xr)0ψ, φ) is
P∞⊗ B(U )-measurable, for every φ ∈ Φ, ψ ∈ Ψ. The above properties and condition
(a) of Definition 4.2.5 imply that {F (r, u, Xr(ω))0ψ : r ∈ [0, t], ω ∈ Ω, u ∈ U } ∈ Λ2,locw (t)
(see Definition 3.2.17) and hence from Theorem 3.2.20 the weak stochastic integral Rt
0
R
UF (r, u, Xr)
0ψM (dr, du) exists for every t ≥ 0 .
Chapter 4. Stochastic Evolution Equations in Duals of Nuclear Spaces 90
Definition 4.2.7. A Ψ0β-valued regular and predictable process X = {Xt}t≥0 is called
a mild solution to (4.10) if
(a) For every t ≥ 0 , for all ψ ∈ Ψ ,
P ω ∈ Ω : Z t 0 S(t − r)0B(r, Xr(ω))[ψ] dr < ∞ = 1. P ω ∈ Ω : Z t 0 Z U qr,u(F (r, u, Xr(ω))0S(t − r)ψ)2µ(du)dr < ∞ = 1. (b) For every t ≥ 0 , P-a.e.
Xt= S(t)0X0+ Z t 0 S(t − r)0B(r, Xr)dr + Z t 0 Z U S(t − r)0F (r, u, Xr)M (dr, du), (4.12) where the first integral at the right-hand side of (4.12) is a Ψ0β-valued regular, {Ft}-adapted process nRt
0 S(t − r)
0B(r, X
r)dr : t ≥ 0
o
such that for all t ≥ 0 and ψ ∈ Ψ , for P-a.e. ω ∈ Ω, Z t 0 S(t − r)0B(r, Xr(ω))dr [ψ] = Z t 0 S(t − r)0B(r, Xr(ω))[ψ]dr, (4.13)
where for each t ≥ 0 , ψ ∈ Ψ , the integral on the right-hand side of (4.13) is the Lebesgue integral of the function 1[0,t](·) S(t − ·)0B(·, X·(ω))[ψ] defined on [0, t]
for P-a.e. ω ∈ Ω. The second integral at the right-hand side of (4.12) is the strong stochastic integral of {1[0,t](r) S(t − r)0F (r, u, Xr(ω)) : r ∈ [0, t], ω ∈ Ω, u ∈ U } .
Proposition 4.2.8. The assumptions (A1)-(A4) together with the conditions (a) of Definition 4.2.7 are sufficient to guarantee the existence of all the integrals in (4.12). Proof. We start with the existence of the process nRt
0S(t − r)
0B(r, X
r)dr : t ≥ 0
o . Fix t ≥ 0 . We need to show that the conditions (1)-(2) of Theorem 4.1.1 are satisfied for the map X : [0, t] × [0, t] × Ω → Ψ0 given by
X(s, r, ω) =1[0,s](r) S(s − r)0B(r, Xr(ω)), for (s, r, ω) ∈ [0, t] × [0, t] × Ω. (4.14)
From the arguments on the proof of Proposition 4.2.6 it follows that the map (r, ω) 7→ B(r, Xr(ω))[ψ] is P∞-measurable, for every ψ ∈ Ψ . Then, for any s ∈ [0, t] , the
continuity of the map r 7→1[0,s](r) S(s − r)ψ , implies that the map (r, ω) 7→ X(s, r, ω) =1[0,s](r) B(r, Xr(ω))[S(s − r)ψ],
is Ps-measurable, for all ψ ∈ Ψ . Hence, X satisfies the condition (1) of Theorem 4.1.1.
On the other hand, the condition (a) of Definition 4.2.7 is exactly the condition (2) of Theorem 4.1.1 for X defined by (4.14). Therefore, Theorem 4.1.1 implies the existence of the process nRt
0 S(t − r)
0B(r, X
r)dr : t ≥ 0
o
satisfying the conditions of Definition 4.2.7.
For the stochastic integral, we have to check that for each t ≥ 0 , the integrand is an element of Λs(t) (Definition 3.3.32).
Fix t ≥ 0 . Let R = {R(r, ω, u)} be given by